Connectivity Of Random Graphs Research Articles

Connectivity of random graphs after centrality-based vertex removal

Abstract Centrality measures aim to indicate who is important in a network. Various notions of ‘being important’ give rise to different centrality measures. In this paper, we study how important the central vertices are for the connectivity structure of the network, by investigating how the removal of the most central vertices affects the number of connected components and the size of the giant component. We use local convergence techniques to identify the limiting number of connected components for locally converging graphs and centrality measures that depend on the vertex’s neighbourhood. For the size of the giant, we prove a general upper bound. For the matching lower bound, we specialise to the case of degree centrality on one of the most popular models in network science, the configuration model, for which we show that removal of the highest-degree vertices destroys the giant most.

Estimation of the Connectivity of Random Graphs Through Q-Learning Techniques

Estimation of the Connectivity of Random Graphs Through Q-Learning Techniques

On the connectivity of random graphs from addable classes

A class A of graphs is called weakly addable (or bridge-addable) if for any G∈A and any two distinct components C1 and C2 in G, any graph that can be obtained by adding an edge between C1 and C2 is also in A. McDiarmid, Steger and Welsh (2006) conjectured in [6] that a graph chosen uniformly at random among all graphs with n vertices in a weakly addable A is connected with probability at least e−1/2+o(1), as n→∞. In this paper we show that the conjecture is true under a stronger assumption. A class G of graphs is called bridge-alterable, if for any G∈G and any bridge e in G, G∈G if and only if G−e∈G. We prove that a graph chosen uniformly at random among all graphs with n vertices in a bridge-alterable G is connected with probability at least e−1/2+o(1), as n→∞.The main tool in our analysis is a tight enumeration result that addresses the number of ways in which a given forest can be complemented to a forest with fewer components.

Connectivity for Random Graphs from a Weighted Bridge-Addable Class

There has been much recent interest in random graphs sampled uniformly from the $n$-vertex graphs in a suitable structured class, such as the class of all planar graphs. Here we consider a general bridge-addable class $\cal A$ of graphs -- if a graph is in $\cal A$ and $u$ and $v$ are vertices in different components then the graph obtained by adding an edge (bridge) between $u$ and $v$ must also be in $\cal A$. Various bounds are known concerning the probability of a random graph from such a class being connected or having many components, sometimes under the additional assumption that bridges can be deleted as well as added. Here we improve or amplify or generalise these bounds (though we do not resolve the main conjecture). For example, we see that the expected number of vertices left when we remove a largest component is less than 2. The generalisation is to consider `weighted' random graphs, sampled from a suitable more general distribution, where the focus is on the bridges.

K-connectivity in random undirected graphs

This paper concerns vertex connectivity in random graphs. We present results bounding the cardinality of the biggest k-block in random graphs of the G n, p model, for any constant value of k. Our results extend the work of Erdös and Rényi and Karp and Tarjan. We prove here that G n, p , with p⩾ c n , has a giant k-block almost surely, for any constant k>0. The distribution of the size of the giant k-block is examined. We provide bounds on this distribution which are very nearly tight. We furthermore prove here that the cardinality of the biggest k-block is greater than n-log n, with probability at least 1−1/( n 2log n), for p⩾ c n and c> k+3.

Connectivity of random graphs

We consider a random field {Xij, i, j = 1, ···, n} where the random variables Xij takes on values 1 or 0. The collection {Xij } can be viewed as a random graph with nodes {1, ···, n} by interpreting X ij = 1 as the existence of an arc emanating from the node i to the node j. Such a representation will enable us to study ordered and unordered graphs, being also the general representation of a random graph. In this note the probability that the graph is connected is computed under the condition that Σ iXki=l for k = 1, · ··, n. This result extends Ross's recent theorems on connectivity of random graphs.

Connectivity of random graphs

We consider a random field {Xij, i, j = 1, ···, n} where the random variables Xij takes on values 1 or 0. The collection {Xij} can be viewed as a random graph with nodes {1, ···, n} by interpreting Xij = 1 as the existence of an arc emanating from the node i to the node j. Such a representation will enable us to study ordered and unordered graphs, being also the general representation of a random graph. In this note the probability that the graph is connected is computed under the condition that ΣiXki=l for k = 1, · ··, n. This result extends Ross's recent theorems on connectivity of random graphs.

An exact probability distribution on the connectivity of random graphs

Recently, several authors have proposed the use of random graph theory to evaluate the adequacy of cluster analysis results. One such statistic is the minimum number of lines (edges) V needed to connect a random graph. Erdös and Rényi derived asymptotic distributions of V. Schultz and Hubert showed in a Monte Carlo study that the asymptotic approximations are poor for small sample sizes n typically used in data analysis applications. In this paper the exact probability distribution of V is given and the distributions for some values of n are tabulated and compared with existing Monte Carlo approximations.

Data analysis and the connectivity of random graphs

This paper presents empirical information on the adequacy of an asymptotic approximation developed by Erdös and Rényi dealing with the minimum number of lines needed to connect a random graph. It is shown that the approximation is poor for several data analysis applications of the result previously published in the behavioral sciences.